Direct Proof Exploration

Given: If x is an odd number, Prove: then x³ is odd. To prove the statement "If x is an odd integer, then x³ is odd" using a direct proof, we follow these steps: 1. Assume that x is an ____________________. 2. By definition, an odd integer can be expressed as _______________, where k is an integer. 3. Substitute x with 2k + 1 in the expression for _____________________. 4. Expanding (2k + 1)³ using the binomial theorem, we get: ______________________________. 5. Notice that every term in the expanded expression is _______________, as each term contains a factor of 2k. 6. Express the expanded expression as 2m + 1, where m is an ____________________. 7. Therefore, we have shown that x³ can be expressed as an odd integer, as it can be written as __________. 8. Conclude that if x is an odd integer, then x³ is odd, as desired. In this direct proof, we assume the premise that x is an odd integer and then use algebraic manipulations to express x³ as an odd integer. This confirms that the statement holds true based on the given assumption.

Use the method of direct proof to prove the following statement: Suppose x, y ∈ Z. If x and y are odd, then xy is odd.